We examine how the average of double-winding Wilson loops depends on the number of color N in the SU(N) Yang-Mills theory. In the case where the two loops C_1 and C_2 are identical, we derive the exact operator relation which relates the double-winding Wilson loop operator in the fundamental representation to that in the higher dimensional representations depending on N. By taking the average of the relation, we find that the difference-of-areas law for the area law falloff recently claimed for

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... N=2 is excluded for N ≥ 3, provided that the string tension obeys the Casimir scaling for the higher representations. In the case where the two loops are distinct, we argue that the area law follows a novel law (N - 3)A_1/(N-1)+A_2 with A_1 and A_2 (A_1<A_2) being the minimal areas spanned respectively by the loops C_1 and C_2, which is neither sum-of-areas (A_1+A_2) nor difference-of-areas (A_2 - A_1) law when (N≥3). Indeed, this behavior can be confirmed in the two-dimensional SU(N) Yang-Mills theory exactly.